Two Concepts of Plausibility in Default Reasoning

Hans Rott

Department of Philosophy, University of Regensburg,93040 Regensburg, Germany

hans.rott@ur.de

1. Introduction

Friedman and Halpern’s work on plausibility measures (Friedman and Halpern1995, 1996, 1999, 2001, Halpern 1997, 2001, 2003) achieved a beautiful unifi-cation of several approaches to default reasoning. As Friedman and Halpernpoint out, plausibility measures can be seen as a generalization of various othersemantics for default reasoning, such as preferential structures (Shoham; Kraus,Lehmann and Magidor, “KLM” in short), ε-semantics (Adams; Pearl; Geffner),possibilistic structures (Dubois and Prade) and κ-rankings (Spohn; Goldszmidtand Pearl). Friedman and Halpern claim that by focussing on such generalmeasures, one can understand why the KLM-properties were bound to play thecentral role in the literature on default reasoning that they did in fact play.

On such a very general level, Friedman and Halpern use a fundamental propertyfor plausibility measures they call Qualitativeness.

If A, B and C are pairwise disjoint and A < B ∪ C and B < A ∪ C,then A ∪B < C

This condition is very similar to a condition called “Choice” that features cen-trally in the entrenchment-based account to belief revision provided by Rott(1996, 2001, 2003).1 Translated into the language of plausibilities, Choice lookslike this:

A < B ∪ C and B < A ∪ C if and only if A ∪B < C

Now default reasoning and belief revision have sometimes been called “two sidesof the same coin” (Gardenfors 1990), and thus this coincidence does not seemsurprising. However, the property mentioned is very uncommon and slightly

1That A ∪B <FH C implies A <FH B ∪ C follows from other conditions of Friedman andHalpern.

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unintuitive. On closer inspection, there is actually a substantial difference be-tween the set-ups of Friedman and Halpern on the one hand and Rott on theother. Friedman and Halpern’s condition is restricted and thus considerablyweaker than Rott’s, and this restricted condition is used in a different contextthan the unrestricted condition. This paper explores the relationship betweenFriedman-Halpern plausibility measures and relations of “basic entrenchment.”I argue that while basically equivalent, the latter way of structuring things hassome clear advantages over that of Friedman and Halpern.

1. The central condition Choice is stronger and yet more natural than thecentral condition of Qualitativeness of FH.

2. The method of retrieving the plausibility relation from a given defaultreasoning system is better motivated than Qualitativeness, which is es-sentially a result of reverse engineering.2

3. There is a one-to-one correspondence between conditions for plausibilityrelations and conditions for inference operations, not just packages ofconditions that are tied together “somewhat surprisingly.”3

The plan of this paper is as follows. In section 2, we take the first steps to mo-tivate the concept of plausibility. I argue that in thinking about plausibility, itmakes sense to think of the concept of default reasoning as having methodolog-ical priority over that of plausibility, and I recommend taking default reasoningto be based on expectation revision. Section 3 recalls some arrangements ofprinciples of default reasoning that are relevant to our discussion. Section 4presents both Friedman-Halpern plausibility and “basic plausibility” in a waythat facilitates the comparison. It is shown how the latter can be vindicated bythe reconstructive view taken, and that the two kinds of plausibility relations aredifferent. In the short section 5, we show that the two kinds of plausibility canbe applied in the same way. In section 6, we first retrieve Friedman-Halpernplausibilities from default inference relations in a way similar to how basicplausibilities have been retrieved. Then we check whether there is a harmonicmapping between default inference relations and plausibility relations. Thisworks fine for both FH plausibilities and basic plausibilities in one direction. Inthe other direction the mapping succeeds only for basic plausibilities. Section7 shows how one can translate directly between the plausibility languages ofFriedman-Halpern and of Rott. According to this translation (well-behaved)basic plausibility relations are subrelations of (qualitative) Friedman-Halpernrelations, and they both agree on disjoint sets. Section 8 gives a more de-tailed philosophical motivation of basic plausibilities. Section 9 gives a shortconclusion.

2Joseph Halpern, personal communication, July 2011.3These are the words of Friedman and Halpern (1996, p. 1301, 2001, p. 658) and Halpern

(2003, p. 303).

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2. The primacy of default reasoning

Default reasoning is reasoning based on defaults. Defaults are assumptionsabout the normal or typical state of affairs. Like beliefs, defaults are in the mindof the agent. In fact, defaults can be thought of as being similar to beliefs, butthey are weaker. A good term to indicate this relative weakness is expectation(Gardenfors and Makinson 1994). Beliefs are based on (“hard”) information,and information overrides (“soft”) default assumptions. Information does notgo against the agent’s expectations as long as it is consistent with them. Butwhat makes default reasoning interesting is that people have, and have to have,many expectations, and it is a frequent experience that the information peopleget conflicts with their default assumptions.

We need to be careful whether we are talking about sentences or propositions.Default reasoning will captured by a kind of logic, that is an inference relation|∼ relating (single) premises to (single) conclusions. The relata thus are sen-

tences. On the other hand, we shall proceed on the idea that beliefs or, morepertinently, expectations are propositions, where a proposition is what is saidby an assertive sentence. In the present paper, propositions are thought of assets of possible worlds. For the sake of simplicity, we presuppose a finitisticframework in which every set of possible worlds is expressible by a single sen-tence. Notation: For any sentence α, [[α]] denotes the proposition expressedby the sentence α, i.e., the set of possible worlds at which α is true. For anyproposition A, φA denotes a sentence that expresses A. Suppose that such asentence φA has been chosen. Then clearly there are many sentences that ex-press the same proposition, namely those that are logically equivalent with φA.So φA is not uniquely determined. As will become clear soon, however, we willalways work in contexts in which it is irrelevant which representative of the setof sentences expressing A is chosen, so everything will be well-defined. And ofcourse, [[φA]] = A, and φ [[α]] is equivalent with α.

What is the concept of plausibility in the context of default reasoning? The con-cept applies paradigmatically to propositions that are surprising in that theygo against the agent’s expectations. As a limiting case, one can also say thatplausibility applies to propositions that are compatible with, or possible accord-ing to, one’s expectations. Such propositions enjoy maximal plausibility. Theyare all equally plausible and more plausible than anything that contradicts theagent’s expectations. But propositions that go against one’s expectations maydo so in varying degrees. That some proposition A is less plausible than anotherproposition B means that the expectations militating against A are stronger,or better entrenched, than the expectations militating against B. It is easy tofind representatives of such expectations: these are just the complements of thepropositions in question. The representative expectation militating against Ais −A, and the representative expectation militating against B is −B. So apotential piece of information A is less plausible than another potential pieceof information B just in case that the expectation −A is more entrenched thanthe expectation −B, in symbols

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(<BP

and <BE

) A <BPB iff −B <

BE−A

Here, <BP

and <BE

are strict relations of (basic) plausibility and (basic)entrenchment, respectively. The duality (<

BPand <

BE) between “plausibility”

and “entrenchment” is, of course, partly just an exercise in terminological stage-setting. But the terms and the bridge between them has been made use of forquite some time. Entrenchment relations are relations of comparative necessity,mainly over beliefs or expectations (defaults), plausibility relations are relationsof comparative possibility, mainly over disbeliefs or disexpectations.4

I suggest to focus on a fundamental notion of plausibility that can be derivedwhenever a default inference relation is given. This is one of the most importantideological differences between the approach advocated in the present paperand that of Friedman and Halpern. It seems to me that the reconstructionof plausibility or entrenchment in terms of default reasoning can be clearlyunderstood, in fact better than the construction of default reasoning in termsof entrenchment or plausibility.

We said that a proposition is less plausible if the expectations militating againstit are stronger or more entrenched. What is the meaning of entrenchment?Entrenchment is always relative to a belief state or, and this is the interpretationthat is relevant here, to a state of expectations. We use the covering termdoxastic state and represent the core of a doxastic state by a proposition X.A proposition A is an expectation or a default proposition in X, if X ⊆ A. Aproposition A is less entrenched (in a doxastic stateX) than another propositionB just in case the former is withdrawn while the latter is retained if at least oneof them has to be withdrawn (from X). We can formalize this straightforwardidea as follows:

(From.− to <

BE) A <

BEB iff X

.−(A ∩B) 6⊆ A and X.−(A ∩B) ⊆ B

The symbol.− signifies the operation of rational withdrawal of a proposition

from the set of expectations X (or, as many people say, the rational contractionof X by a proposition). Withdrawing at least one of the expectations A and Bjust means, I submit, withdrawing the expectation A∩B. In the principal case,then, saying that A is less entrenched than B just means that B is retained,but A is not retained.

The thesis that withdrawing A∩B just means withdrawing at least one of A andB looks problematic. There are examples in which withdrawing A∩B is clearlydifferent from both withdrawing A and withdrawing B, or any combinationthereof.

4Both entrenchment and plausibility relations may be considered as representing degreesof belief. Compare Rott (2009), with references to the seminal work produced within theframework of Dubois and Prade’s possibility theory.

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Example 1. Suppose we have a plausibility relation ≺ on the worlds in W ={u, v1, v2, w1, w2}, and that the contraction of X = {u} by a proposition isobtained by joining X with the most plausible worlds not satisfying this propo-sition. Let P = {v1, w2} and Q = {v2, w1}. Suppose that u is more plausiblethan all the other worlds, i.e., vi ≺ u and wi ≺ u for all i = 1, 2, and thatv2 ≺ v1 and w2 ≺ w1, and that no other pair of worlds is related by ≺. Theresult of giving up a proposition A in X is the proposition obtained by takingthe union of X with the set of most plausible worlds at which A is not true.So, in our example, the result of giving up −P is X

.−−P = {u, v1, w2} and theresult of giving up −Q is X

.− −Q = {u, v2, w1}. But the result of giving up−P ∩ −Q is X

.−(−P ∩ −Q) = {u, v1, w1} which is evidently not equal to eitherone of the former sets. Nor can it be obtained by their intersection or anyother Boolean combination of them. So one might be tempted to think that“withdrawing −P ∩ −Q” cannot mean “withdrawing at least one of −P and−Q.” But this would be a mistake. It is simply not the case that “withdrawingat least one of two propositions A and B” means the same as “withdrawingA”-or-“withdrawing B”-or-“withdrawing A and withdrawing B”.5

For default reasoning, it is not enough to withdraw certain propositions. Sucha step is made just in order to make possible the consistent expansion of theexpectation set by another proposition—namely the premise of the default in-ference. Using the so-called “Harper identity” (Gardenfors 1988, p. 70) linkingbelief/expectation contraction and belief/expectation revision, we get:

(From ∗ to <BE

) A <BEB iff X ∗ −(A ∩B) 6⊆ A and X ∗ −(A ∩B) ⊆ B

Here the symbol ∗ signifies the operation of rational revision of the expectationsby a proposition. The condition X ⊆ B, which is also required on the right-hand side by the Harper identity, is implied by X ∗ −(A ∩B) ⊆ B, given somestandard assumptions concerning consistent revisions.

Let us interpret default reasoning as expectation revision (in the style of Gardenforsand Makinson 1994) and define

( |∼ and ∗) α |∼β (at a doxastic state X) iff X ∗ [[α]] ⊆ [[β]]

Putting the above pieces together yields:

(From |∼ to <BE

) A <BEB iff ¬(φA ∧ φB) |6∼φA and ¬(φA ∧ φB) |∼φB

and5Alchourron, Gardenfors and Makinson (1985, pp. 525–526) discuss a postulate called

“Ventilation” according to which “withdrawing A∩B” means exactly the latter. But this is avery strong postulate that has no counterpart in the core of default reasoning as understoodby Friedman and Halpern. As AGM show, Ventilation implies Rational monotony (see section3 below).

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(From |∼ to <BP

) A <BPB iff φA ∨ φB |∼¬φA and φA ∨ φB |6∼¬φB

Thus, saying that A is less plausible than B just means that ¬φA, but not ¬φB,is defeasibly derivable if one is provided with the sole premise φA ∨ φB.6 Thisgives a very clear meaning to the plausibility relation used, one that is directlyconnected with its use in belief revision and default reasoning. One of the goalsof the present paper is to argue that this meaning of the term “plausibility” isthe most adequate one.

Let us use the notation PB( |∼ ) for the (basic) plausibility relation <BP

derivedfrom a given inference relation |∼ by way of (From |∼ to <

BP).7

3. Postulates for default inference relations

An inference relation is a set of elements of the form α |∼β that is closed undera certain set of conditions. Friedman and Halpern emphasize that the followingset of conditions for default reasoning has proved to be of particular relevance.This “core” defines what has been called preferential reasoning since the seminalpaper of Kraus, Lehmann and Magidor (1990).

(LLE) If α a` β, then α |∼ γ iff β |∼ γ (Left logical equivalence)

(RW) If α ` β and γ |∼α, then γ |∼β (Right weakening)

(REF) α |∼α (Reflexivity)

(AND) If α |∼β and α |∼ γ, then α |∼β ∧ γ (And)

(OR) If α |∼ γ and β |∼ γ, then α ∨ β |∼ γ (Or)

(CM) If α |∼β and α |∼ γ, then α ∧ γ |∼β (Cumulative monotony)

In Friedman and Halpern’s presentation, these six postulates come in two pack-ages of three elements each: The general case is defined by the set consisting of(RW), (LLE) and (REF), the qualitative case adds the set of (And), (Or) and(CM). From a belief revision point of view, it is the collection of the first fourconditions, (LLE), (RW), (REF) and (AND) that has been called basic. Theseconditions correspond to the basic postulates of AGM (Alchourron, Gardenforsand Makinson 1985). While Friedman and Halpern’s general case is designedto be open to probabilistic interpretations, AGM have been more qualitative tobegin with. (OR) corresponds to the first, and (CM) is essentially a weakening

6Or if one just assumes φA ∨ φB as the sole premise. The same idea, by the way, alreadyappears (with the ordering reversed) in Lewis (1973, p. 54). The point of the present paper isthat this idea can be put to work in much weaker settings than the ones envisaged by Lewis.

7Essentially the same method of deriving a plausibility relation from a default inferencerelation was used by Lehmann and Magidor (1992, pp. 31, 45) and by Freund (1993, 1998).

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of the second supplementary postulate of AGM.8 Friedman and Halpern do not(and, I think, cannot) take apart the two packages in their analysis. We willbe able handle the cases one by one in the present paper (except for the verybasic conditions (REF) and (LLE)).

We also mention two more conditions.

(CP) If α |∼⊥, then α ` ⊥ (Consistency preservation)

(RM) If α |∼β and α |6∼¬γ, then α ∧ γ |∼β (Rational monotony)

The well-known condition of Consistency Preservation, (CP), makes it a loteasier to deal with some limiting cases, but it appears to be very strong froma philosophical point of view. So it would by preferable to work without (CP),were it not for the fact that this would incur considerable technical costs.9 Wewill make use of it in the following presentation.

Rational monotony, (RM), is a condition made prominent by Lehmann andMagidor (1992), its counterpart for belief revision has been an essential part ofthe postulates of Alchourron, Gardenfors and Makinson (1985). However, inmany contexts one may want to reject it as being excessively strong. I agreewith Friedman and Halpern (and many others) that it should not be countedas belonging to the “core properties of default reasoning”.

4. Two kinds of plausibility relations

4.1. Friedman-Halpern plausibility relations

Friedman and Halpern use the following set of axioms for their plausibilityrelations <

FH:

(FH0a) Not A <FHA (Irreflexivity)

(FH0b) If A <FHB and B <

FHC, then A <

FHC (Transitivity)

8The supplementary postulates are labelled postulates number 7 and 8 in the usual AGMnumbering. The second supplementary postulate of AGM corresponds to Rational monotonywhich is very strong (see below). Until some time in the 1990s, the research in nonmonotonicreasoning recognized finer distinctions than belief revision research.

9One could replace (CP) by the weaker conditions

(⊥Cond) If α ∧ β |∼⊥, then α |∼¬β and

(⊥CM) If α |∼⊥, then α ∧ β |∼⊥(⊥Cond) and (⊥CM) are very special cases of (Cond), If α ∧ β |∼ γ, then α |∼β→γ, and(CM). Taken together, they are still a lot weaker than (CP). (Note that(⊥OR), If α |∼⊥and β |∼⊥, then α ∨ β |∼⊥ would not be strong enough as a replacement of (⊥Cond). Itis not (relatively) equivalent with (⊥Cond), even though (OR) is (relatively) equivalent with(Cond).) – Friedman and Halpern’s condition (FH3) mentioned below, and also the conditions(E∅1) and (E∅2) discussed in Rott (2001, p. 235), are examples of similar technical conditionsthat are, I think, of little intrinsic interest.

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(FH1≤) If A ⊆ B, then A ≤FHB (FH-Dominance)

(FH2) If A, B and C are pairwise disjoint and A <FHB∪C and B <

FHA∪C,

then A ∪B <FHC (Qualitativeness)

(FH3) If ∅ <FHA ∪B, then either ∅ <

FHA or ∅ <

FHB (Bottom)

The most general concept of a plausibility relation in the sense of Friedman andHalpern is defined by (FH0a), (FH0b) and (FH1≤).10 Much of the beauty ofFriedman and Halpern’s approach lies in the fact that the elementary conditions(FH0a), (FH0b) and (FH1≤) are suitable not only for the ”qualitative” kindsof reasoning they focus on, but are at the same time valid for probabilisticapproaches. We are interested in working with the strict relation < ratherwith a non-strict relation ≤.11 So I suggest to express the content of the non-strict relationship “A ≤

FHB” by the condition that for all propositions C, if

C <FHA, then C <

FHB, and if B <

FHC, then A <

FHC. The FH condition of

Dominance is thus transformed into the following variant:

(FH1) If A ⊆ B and C <FHA, then C <

FHB;

if A ⊆ B and B <FHC, then A <

FHC (FH-Dominance)

Despite its apparent complexity, this is a very natural condition. A smallerproposition A is at most as plausible as a larger proposition B. Thus everythingdominated by the former should be dominated by the latter, and everythingdominating the latter should dominate the former.

Adding (FH2) and (FH3) gives what Friedman and Halpern call qualitativeplausibility relations. To the best of my knowledge, the Qualitativeness condi-tion was first mentioned by Friedman and Halpern (1995, p. 182) and Duboisand Prade (1995, p. 155).12 (FH2) is the most interesting condition. There isin fact some magic to it. Friedman and Halpern discovered it when looking fora condition that is tailor-made for guaranteeing that the (AND) rule is satisfiedin default reasoning. But then, “somewhat surprisingly”,13 it turned out that(FH2) at the same time yields cumulative monotony, (CM), and the principalcase of the (OR) rule. On the one hand, this can be viewed as good news(and this is how Friedman and Halpern themselves have seen it): Only a singlecondition is needed to get essentially three central conditions of default reason-ing. On the other hand, this may also be regarded as slightly unsettling: How

10They actually use plausibility measures, i.e., mappings that assign a member of a partiallyordered set to each element of the domain. I will not attend to this difference in the presentpaper.

11Why? Because there is no “local” way of distinguishing between plausibility ties andplausibility incomparabilities. This can be seen from the basic idea of (From |∼ to <BP ): Ifthe agent’s premise is φA∨φB and she then disbelieves neither A nor B, this may be so eitherbecause A and B are tied or because they are incomparable in terms of plausibility. Also cf.Rott (1992, p. 50).

12Also see Dubois, Fargier and Prade (2004, pp. 34–35).13See footnote 3.

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exactly does this condition manage to achieve three tasks at the same time?Wouldn’t it be more perspicuous if we had a modular approach that is capableof accounting for the three central conditions (AND), (CM) and (OR) one byone? This is what I want to offer with the present paper.

4.2. Basic plausibility relations

Basic plausibility relations <BP

are the duals to basic entrenchment relationsas presented in Rott (2003). They are relations between propositions, too.14

They are defined by the following four conditions:

(BP0) Not A <BPA (Irreflexivity)

(BP1) If A 6= ∅, then ∅ <BPA (Minimality)

(BP2) If A ∪B <BPC, then A <

BPB ∪ C (Choice easy)

(BP3) If A <BPB ∪ C and B <

BPA ∪ C, then A ∪B <

BPC (Choice hard)

Minimality (BP1) corresponds to (Consistency preservation) in default reason-ing. The conjunction of the conditions (BP2) and (BP3), or rather the counter-part of this conjunction for entrenchment relations, was called “Entrenchmentand choice” in (Rott 2001, p. 233) and simply “Choice” in (Rott 2003, p. 266).It played a crucial role in the choice-theoretic reinterpretation of belief revisionand nonmonotonic reasoning offered in Rott (2001); we will briefly turn to thistopic in section 8. (BP3) is an unrestricted variant of Friedman and Halpern’scondition of Qualitativeness (FH3). This point was already mentioned in theintroduction.15

Now we take down some of the properties of basic plausibility relations:

Lemma 1. Basic plausibility relations satisfy the following conditions.

(BP-i) If A <BPB, then not B <

BPA (Asymmetry)

(BP-ii) A <BPB iff A <

BPA ∪B (Join right)

(BP-iii) A <BPB iff A <

BP−A ∩B (Meet right)

(BP-iv) If A <BPB, then B 6⊆ A (GM-Dominance)

14Basic entrenchment relations were originally presented as relations between sentencesrather than propositions. If we wanted to do the same here for the dual notion of plausibility,we would need a condition of Extensionality, like this one:

If α a` β, then: α <BP γ iff β <BP γ, and γ <BP α iff γ <BP β15However, somewhat confusingly, Halpern (1997, p. 5) uses the term “qualitative” for

relations that satisfy the unrestricted condition (Choice hard). Since this usage was notmaintained in (Friedman and) Halpern’s later work, I will ignore it from now on.

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(BP-v) If A <BPC and B <

BPC and A ∪B ⊆ C, then A ∪B <

BPC

(Weak join left)

(BP-vi) If A <BPB and A ∩ −B ⊆ C ⊆ A, then C <

BPB

(Weak continuing down)

(BP-vii) If A <BPB and B ⊆ C ⊆ A ∪B, then A <

BPC

(Weak continuing up)16

Proof. (Asymmetry) Suppose for reductio that A <BPB and B <

BPA, i.e.,

A ∪ A <BPB and B ∪ B <

BPA. By (Choice easy), we get A <

BPA ∪ B and

B <BPA∪B. This also means that A <

BPB∪ (A∪B) and B <

BPA∪ (A∪B).

By (Choice hard), it follows that A∪B <BPA∪B, contradicting (Irreflexivity).

(Join right) Let A <BPB. This means that A ∪ A <

BPB. By (Choice easy)

and (Choice hard), this is equivalent with A <BPA ∪B.

(Meet right) Let A <BPB. By (Join right), this is true just in case that

A <BPA∪B. Since A∪(−A∪B) = A∪B, this means that A <

BPA∪(−A∪B).

By (Choice easy) and (Choice hard), this is true just in case A <BP−A ∪B.

(GM-Dominance) Suppose for reductio that A <BPB and B ⊆ A. Since

A = A ∪ B, we get A ∪ B <BPB. So by (Join right), A ∪ B <

BP(A ∪ B) ∪ B,

contradicting (Irreflexivity).(Weak join left) Let A <

BPC, B <

BPC and A ∪ B ⊆ C. Since A and B are

subsets of C, we have A <BPB ∪ C and B <

BPA ∪ C, so by (Choice hard)

A ∪B <BPC.

(Weak continuing down) Let A <BPB and A∩−B ⊆ C ⊆ A. Since A∪C = A,

A ∪C <BPB. So by (Choice easy), C <

BPA ∪B. But A ∪B = C ∪B, so this

means that C <BPC ∪B. Thus by (Choice hard) C <

BPB.

(Weak continuing up) Let A <BPB and B ⊆ C ⊆ A∪B. By (Choice easy), we

get A <BPA∪B. Since A∪C = A∪B, we get A <

BPA∪C. Thus by (Choice

hard) A <BPC. QED

As we have seen, basic plausibility functions have quite a number of nice proper-ties. Still they may be surprisingly ill-behaved. They are not in general acyclic,and thus, given irreflexivity, they are not in general transitive.17 This defect canbe remedied in an instructive way if the following two conditions are employed:

(BP4) If A <BPB, then A <

BPB ∪ C (Continuing up)

(BP5) If A ∪ C <BPB, then A <

BPB (Continuing down)

Let us call basic plausibility relations <BP

that satisfy (Continuing up) and(Continuing down) well-behaved.

16This condition could be strengthened toIf A <BP B and −A ∩B ⊆ C ⊆ A ∪B, then A <BP C

with essentially the same proof, but then it would not be a weakened form of (Continuing up)(see below) any more.

17A counterexample for the dual notion of basic entrenchment is given in Rott (2003, p.268, footnote 20).

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Analogues of (BP4) and (BP5) have been known under the names “Continu-ing up” and “Continuing down” since Alchourron & Makinson (1985). Takentogether, they are essentially identical with (our rendering of) the Friedman-Halpern condition of Dominance (FH1), which we have called a natural condi-tion. (BP4) and (BP5) immediately imply (Choice easy), and one half of eachof (BP-ii) and (BP-iii). But they also give rise to substantially new propertieswhich we collect in the next lemma.

Lemma 2. (a) Well-behaved basic plausibility relations satisfy the followingconditions.

(BP-viii) If A <BPB and B <

BPC, then A <

BPC (Transitivity)

(BP-ix) If A <BPC and B <

BPC, then A ∪B <

BPC (Join left)

(BP-x) If A <BPC and B <

BPD, then A ∪B <

BPC ∪D (Join left&right)

(b) An alternative axiomatization of well-behaved basic plausibility relationscan be given by Irreflexivity (BP0), Minimality (BP1), (Continuing up) (BP4),(Continuing down) (BP5), Join right (BP-ii) and Join left (BP-ix).18

Proof. (a) (BP-viii), transitivity: Let A <BPB and B <

BPC. Then, by

(Continuing up), A <BPB ∪C and B <

BPA ∪C. By (Choice hard), it follows

that A ∪B <BPC. So by (Continuing down), A <

BPC.19

(BP-ix): Let A <BPC and B <

BPC. By (Continuing up), A <

BPB ∪ C and

B <BPA ∪ C. So by (Choice hard), A ∪B <

BPC.

(BP-x): Let A <BPC and B <

BPD. By (Continuing up), A <

BPB ∪ C ∪ D

and B <BPA ∪ C ∪D. So by (Choice hard), A ∪B <

BPC ∪D.

(b) As already noted, (Choice easy) follows from (Continuing up) and (Con-tinuing down). It remains to show that (Choice hard) is valid. Suppose thatA <

BPB ∪ C and B <

BPA ∪ C. Then, by (Continuing up), A <

BPA ∪ B ∪ C

and B <BPA ∪B ∪ C. So, by (BP-ix), A ∪B <

BPA ∪B ∪ C, and by (BP-ii),

A ∪B <BPC. QED

The last condition of interest is Modularity (also known as Virtual connectivityor Negative transitivity):

18This in fact is essentially the axiomatization given by Freund (1993, pp. 237–238). It isessentially dual the to axiomatization of “generalized epistemic entrenchment” given in Rott(1992, p. 55).

19It is easy to show that the following condition corresponds precisely to the transitivity of<BP

(+) If α ∨ β |∼¬α and β ∨ γ |∼¬β, then α ∨ γ |∼¬αThis condition is valid in preferential reasoning. Proof: From α ∨ β |∼¬α and ¬α ∧ γ |∼¬α itfollows by (OR) that α ∨ β ∨ γ |∼¬α. Similarly, we get from β ∨ γ |∼¬β that α ∨ β ∨ γ |∼¬β.By (REF), (AND) and (RW), we get α ∨ β ∨ γ |∼α ∨ γ. So by (CM) and (LLE), α ∨ γ |∼¬α.Since apparently (+) cannot be nicely simplified and is not of any intrinsic interest, it will notbe mentioned here any more.

11

(BP6) If A <BPB, then A <

BPC or C <

BPB (Modularity)

It is well-known in the theory of belief revision and nonmonotonic reasoningthat this condition corresponds to Rational Monotony.

4.3. The properties of basic plausibility vindicated

In the light of our retrieval idea embodied in (From |∼ to <BP

), it turns outthat the properties of basic entrenchment can be nicely vindicated.

Lemma 3. Let the basic plausibility relation <BP

be derived from a giveninference relation |∼ by way of (From |∼ to <

BP), that is, let <

BP= PB( |∼ ).

Then

(i) (Irreflexivity) is trivially satisfied,

(ii) (Choice easy) follows from (RW),

(iii) (Choice hard) follows from (RW) and (AND),

(iv) (Continuing up) follows from (LLE), (RW), (REF), (AND), (⊥CM) and(OR),

(v) (Continuing down) follows from (LLE), (RW), (AND), (⊥Cond) and(CM).

Proof. Suppose that <BP

is generated from |∼ by (From |∼ to <BP

). Becauseof (LLE) and (RW), <

BPis well-defined. Since (LLE) is not contested we will

use it without mentioning it explicitly.(i) (Irreflexivity) is trivial.(ii) (Choice easy) Let A ∪ B <

BPC. We want to show that A <

BPB ∪ C and

B <BPA ∪ C. Using (From |∼ to <

BP), the supposition gives us

(φA ∨ φB) ∨ φC |∼¬(φA ∨ φB) and (φA ∨ φB) ∨ φC |6∼¬φC .We need to show thatφA ∨ (φB ∨ φC) |∼¬φA and φA ∨ (φB ∨ φC) |6∼¬(φB ∨ φC),as well asφB ∨ (φA ∨ φC) |∼¬φB and φB ∨ (φA ∨ φC) |6∼¬(φA ∨ φC).This, too, follows straightforwardly from the supposition by repeated applica-tions of (LLE) and (RW).(iii) (Choice hard) Let A <

BPB ∪ C and B <

BPA ∪ C. We want to show that

A ∪B <BPC. Using (From |∼ to <

BP), the supposition gives us

φA ∨ φB ∨ φC |∼¬φA and φA ∨ φB ∨ φC |6∼¬(φB ∨ φC),as well asφA ∨ φB ∨ φC |∼¬φB and φA ∨ φB ∨ φC |6∼¬(φA ∨ φC).We need to show thatφA ∨ φB ∨ φγ |∼¬(φA ∨ φB) and φA ∨ φB ∨ φγ |6∼¬φC .

12

This follows straightforwardly from the supposition by repeated applications of(AND) and (RW).(iv) (Continuing up) Let A <

BPB. We want to show that A <

BPB ∪C. Using

(From |∼ to <BP

), the supposition gives usφA ∨ φB |∼¬φA and φA ∨ φB |6∼¬φB.We need to show thatφA ∨ φB ∨ φC |∼¬φA and φA ∨ φB ∨ φC |6∼¬(φB ∨ φC).By (REF) and (RW), we have that¬φA ∧ φC |∼¬φA.So by (OR), we get(φA ∨ φB) ∨ (¬φA ∧ φC) |∼¬φA, which simplifies to φA ∨ φB ∨ φC |∼¬φA.Now suppose for reductio that alsoφA ∨ φB ∨ φC |∼¬(φB ∨ φC).Then by (AND) and (RW), φA ∨ φB ∨ φC |∼⊥.But then by (⊥CM) and (LLE), φA ∨ φB |∼⊥and by (RW) φA ∨ φB |∼¬φB,and we have a contradiction.(v) (Continuing down) Let A ∪ C <

BPB. We want to show that A <

BPB.

Using (From |∼ to <BP

), the supposition gives usφA ∨ φB ∨ φC |∼¬(φA ∨ φC) and φA ∨ φB ∨ φC |6∼¬φB.We need to show thatφA ∨ φB |∼¬φA and φA ∨ φB |6∼¬φB.From the supposition, we get by two different applications of (RW) that bothφA ∨ φB ∨ φC |∼¬φA andφA ∨ φB ∨ φC |∼φA ∨ φB ∨ ¬φC .So by (CM) and (LLE) φA ∨ φB |∼¬φA.Now suppose for reductio that alsoφA ∨ φB |∼¬φB.Then by (AND) and (RW), φA ∨ φB |∼⊥.By (LLE) and (⊥Cond), we get(φA ∨ φB ∨ φC) ∧ (φA ∨ φB) |∼⊥ and φA ∨ φB ∨ φC |∼¬(φA ∨ φB).So finally, by (RW)φA ∨ φB ∨ φC |∼¬φB,and we have a contradiction. QED

Lemma 3 does not give a full 1-1 mapping between the properties of the defaultinference relation |∼ and the properties of the derived plausibility relation<

BP= PB( |∼ ). But even in the more problematic cases (iv) and (v), it seems

justified to call (OR) the property that essentially yields (Continuing up) and(CM) the property that essentially yields (Continuing down).

4.4. Basic plausibility relations are different from FH plausibility relations

4.4.1. Not all basic plausibility relations are FH plausibility relations

As we have pointed out, basic plausibility relations need not be transitive.

13

But all basic plausibility relations that satisfy (Continuing up) and (Continuingdown) are FH plausibility relations. (FH0a) is (BP0), (FH0b) is (BP-viii),(FH1) is the conjunction of (Continuing up) and (Continuing down), (FH2) isa restricted form of (Choice hard), and (FH3) follows from (BP0) and (BP1).

4.4.2. Not all FH plausibility relations are basic plausibility relations

It turns out that (Choice) is stronger than (Qualitativeness), even in the pres-ence of Friedman and Halpern’s other conditions One might have suspected thatFriedman and Halpern’s restricted condition of (Qualitativeness) is sufficient toimply the (Choice) condition, if the more basic properties of FH plausibilitiesare satisfied as well. This, however, is not true. Joseph Halpern has come upwith the following counterexample.

Example 2.20 Let W = {u, v, w} and assign plausibility values to the subsets ofW as follows: plaus(∅) = 0, plaus({u}) = 2, plaus({v}) = 1, plaus({w}) = 1,plaus({u, v}) = 2, plaus({u,w}) = 2, plaus({v, w}) = 1 and plaus({u, v, w}) =3. The plausibility relation < between the subsets of W is supposed to derivefrom these numbers in the obvious way: A < B iff plaus(A) < plaus(B). Theonly non-trivial comparison for Qualitativeness concerns the three disjoint sets{u}, {v} and {w}, and we can indeed verify that

{w} < {v} ∪ {u} and {v} < {w} ∪ {u} taken together imply {v} ∪ {w} < {u}

(This is just 1 < 2 everywhere.) Thus the condition Qualitativeness is sat-isfied in this example. However, for sets that are not disjoint, the analogousimplication does not longer hold. We find that

{v, w} < {u,w} ∪ {u, v} and {u,w} < {v, w} ∪ {u, v},but not {u,w} ∪ {v, w} < {u, v}.

(This is 1 < 3 and 2 < 3 on the left-hand side, but 3 6< 2 on the right-handside.) Thus the condition Choice does not hold here.

The counterexample shows that Qualitativeness does not imply Choice, even inthe presence of Friedman and Halpern’s remaining conditions (FH0a), (FH0b),(FH1≤) and (FH3).

5. Using plausibility relations for default reasoning

FH use the following condition for the construction of a default inference relationon the basis of their plausibility relations21

20Personal communication, July 2011. Dubois, Fargier and Prade (2004, pp. 34) remarkthat Choice is much stronger than Qualitativeness, but they do not present a counterexample.

21See Friedman and Halpern (2001, p. 655) and Halpern (2003, p. 302).

14

(From <FH

to |∼ ) α |∼β iff [[α ∧ ¬β]] <FH

[[α ∧ β]] or ∅ 6<FH

[[α]]

The condition for basic plausibility uses the dual of the entrenchment-basedmethod:22

(From <BP

to |∼ ) α |∼β iff [[α ∧ ¬β]] <BP

[[α]] or [[α]] = ∅.

We seem to have different suggestions here. However, given (BP-iii), the maincondition [[α ∧ ¬β]] <

BP[[α]] is equivalent with [[α ∧ ¬β]] <

BP[[α ∧ β]] .23 The

limiting cases are equivalent given (Minimality). We want to endorse (Mini-mality) for the sake of simplicity. So there is no real difference here, and we canuse the following equation generically, for both kinds of plausibility relations.

(From < to |∼ ) α |∼β iff [[α ∧ ¬β]] < [[α ∧ β]] or [[α]] = ∅.

Let us use the notation I(<) for the default inference relation |∼ based on agiven plausibility relation < by way of (From < to |∼ ).

Lemma 4. Let |∼ be generated from < by (From < to |∼ ). Then

|∼ satisfies (LLE).

If < satisfies (Irreflexivity), then |∼ satisfies (REF).

If < satisfies (Choice easy), then |∼ satisfies (RW).

If < satisfies (Choice easy) and (Choice hard), then |∼ satisfies (AND).

If < satisfies (Choice easy), (Choice hard) and (Continuing up), then |∼ satis-fies (OR).

If < satisfies (Choice easy), (Choice hard) and (Continuing down), then |∼satisfies (CM).

Proof. (LLE) Let α a` β and α |∼ γ, that is, [[α ∧ ¬γ]] < [[α ∧ γ]] or [[α]] = ∅.For β |∼ γ we have to show that [[β ∧ ¬γ]] < [[β ∧ γ]] or [[β]] = ∅. But thisis immediate since α a` β implies that [[β]] = [[α]] , [[β ∧ γ]] = [[α ∧ γ]] and[[β ∧ ¬γ]] = [[α ∧ ¬γ]] .

(REF) For α |∼α we need to show that [[α ∧ ¬α]] < [[α ∧ α]] or [[α]] = ∅. Butthe former reduces to ∅ < [[α]] . This is fulfilled if [[α]] 6= ∅, by (BP1).

(RW) Let α |∼β and β ` γ. Then [[α ∧ ¬β]] < [[α ∧ β]] or [[α]] = ∅. If thelatter, we have α |∼ γ by definition. So assume the former. We need to show

22Used in Rott (2001, pp. 254, 264) and in Rott (2003, p. 264).23This equivalence is not valid for FH plausibility relations.

15

that [[α ∧ ¬γ]] < [[α ∧ γ]] . Since [[α ∧ ¬β]] = [[α ∧ ¬β ∧ γ]] ∪ [[α ∧ ¬γ]] , wehave [[α ∧ ¬β ∧ γ]] ∪ [[α ∧ ¬γ]] < [[α ∧ β]] . So by (Choice easy), [[α ∧ ¬γ]] <[[α ∧ ¬β ∧ γ]] ∪ [[α ∧ β]] . Equivalently, [[α ∧ ¬γ]] < [[α ∧ γ]] , and this is what wewanted to show.

(AND) Let α |∼β and α |∼ γ. Then [[α∧¬β]] < [[α∧β]] and [[α∧¬γ]] < [[α∧γ]] ,or [[α]] = ∅. If the latter, we have α |∼β ∧ γ by definition. So suppose theformer. We have to show that [[α∧¬(β ∧ γ)]] < [[α∧ β ∧ γ]] . By (Choice easy),[[α∧¬β]] < [[α]] and [[α∧¬γ]] < [[α]] . So by (BP-v), [[α∧¬β]] ∪ [[α∧¬γ]] < [[α]] ,or equivalently, [[α ∧ ¬(β ∧ γ)]] < [[α]] . By (BP-iii), [[α ∧ ¬(β ∧ γ)]] < −( [[α ∧¬(β ∧ γ)]] ) ∩ [[α]] , or equivalently, [[α ∧ ¬(β ∧ γ)]] < [[α ∧ β ∧ γ]] , which is whatwe wanted to show. (Note that (BP-iii) and (BP-v) follow from (Choice easy)and (Choice hard).)

(OR) Let α |∼ γ and β |∼ γ. Then [[α∧¬γ]] < [[α∧γ]] or [[α]] = ∅, and [[β∧¬γ]] <[[β∧γ]] or [[β]] = ∅. If [[α]] = ∅ or [[β]] = ∅, then β or α is logically equivalent withα ∨ β, and we get α ∨ β |∼ γ by (LLE). So suppose that [[α]] 6= ∅ and [[β]] 6= ∅.We have to show that [[(α∨β)∧¬γ)]] < [[(α∨β)∧γ]] . By (Continuing up), we get[[α∧¬γ]] < [[β∧¬γ]] ∪ [[α∧γ]] ∪ [[β∧γ]] and [[β∧¬γ]] < [[α∧¬γ]] ∪ [[α∧γ]] ∪ [[β∧γ]] .So by (Choice hard), [[α ∧ ¬γ]] ∪ [[β ∧ ¬γ]] < [[α ∧ γ]] ∪ [[β ∧ γ]] , and this isequivalent with [[(α ∨ β) ∧ ¬γ)]] < [[(α ∨ β) ∧ γ]] , which is what we wanted toshow.

(CM) Let α |∼β and α |∼ γ. Then [[α∧¬β]] < [[α∧β]] and [[α∧¬γ]] < [[α∧γ]] , or[[α]] = ∅. If the latter, we have [[α∧ γ]] = ∅, and we get α ∧ γ |∼β by definition.So suppose the former. We have to show that [[α ∧¬β ∧ γ]] < [[α ∧ β ∧ γ]] . Wefirst copy the proof of (AND) and derive [[α ∧ ¬(β ∧ γ)]] < [[α ∧ β ∧ γ]] . Nowwe note that [[α∧¬β ∧ γ]] ⊆ [[α∧¬(β ∧ γ)]] , apply (Continuing down) and get[[α ∧ ¬β ∧ γ]] < [[α ∧ β ∧ γ]] , as desired. QED

Like Lemma 3, Lemma 4 does not give a full 1-1 mapping between the propertiesof plausibility relations and properties of default inference relation. But again,even in the last two cases which appear to be most problematic, (Continuingup) is quite definitely the property corresponding to (OR), and (Continuingdown) is the property corresponding to (CM).

6. Plausibility-based default reasoning

6.1. FH plausibilities retrieved from default reasoning

In section 2, we mentioned that it has been a central idea of basic entrenchmentto interpret it as retrieved from a belief/expectation withdrawal operation. Weshowed how basic plausibility can equally well be retrieved from a given defaultinference relation and obtained the condition (From |∼ to <

BP). There is

a comparable move in Friedman and Halpern’s work (2001, pp. 657, 669–70,Lemma 4.1) that starts from sets of defaults satisfying the rules of preferentialreasoning.

16

(From |∼ to ≤FH

) A ≤FHB iff φA ∨ φB |∼φB

This retrieval method was used by Kraus, Lehmann and Magidor (1990, Def-inition 5.9),24 who observed that the relation ≤

FHthus defined is transitive,

provided that |∼ is a preferential system (Kraus et al. 1990, Lemma 5.5). Thisobservation was used by Friedman and Halpern (2001, Proof of Lemma 4.1).It should be noted, however, that this was a mainly technical matter for Fried-man and Halpern, while we have obtained (From |∼ to <

BP) as a result of our

attempt to gain an intuitive understanding of the concept of plausibility.

As already pointed out on page 8, we are interested in working with a strictrelation <

FHrather than with the non-strict relation ≤

FH. It is natural

to assume that we obtain the appropriate strict relation just by taking theasymmetric part of Friedman and Halpern’s non-strict relation ≤

FH. This is

captured by the following definition:

(From |∼ to <FH

) A <FHB iff φA ∨ φB |∼φB and φA ∨ φB |6∼φA

Let us use the notation PFH( |∼ ) for the plausibility relation <FH

so derivedfrom a given inference relation |∼ .

What is the relation between PFH( |∼ ) and PB( |∼ )? Provided that |∼ satisfies(REF), (AND) and (RW), φA ∨ φB |∼¬φA and φA ∨ φB |6∼¬φB taken togetherimply the conjunction of φA ∨ φB |∼φB and φA ∨ φB |6∼φA. So <

BP= PB( |∼ )

is a subrelation of <FH

= PFH( |∼ ).25

But the two relations are in general different. The recipes (From |∼ to <BP

)and (From |∼ to <

FH) come apart just in case φA ∨ φB |∼φB, φA ∨ φB |6∼φA

and φA ∨ φB |6∼¬φA. This is a consistent state of affairs. For instance, ifthe plausibility of a proposition is “measured by” the plausibility of its mostplausible worlds (as is common in many semantics), then the two relations aredivergent if the set of most plausible worlds in A ∪ B is contained in B, butintersects both A and −A. In such cases, we have A <

FHB but not A <

BPB.26

Here is an example:

Example 3. Consider the plausibility relation between three worlds u, v and wwith u being more plausible than both v and w, and v and w being unrelated.Suppose that the inference operation |∼ is based on this pre-ordering: α |∼βholds just in case all the maximally plausible worlds in [[α]] are in [[β]] . Let Abe {v} and B be {v, w}. Here we have in fact φA ∨ φB |∼φB, φA ∨ φB |6∼φAand φA ∨ φB |6∼¬φA. Thus A <

FHB but not A <

BPB.

24As mentioned in footnote 7, Lehmann and Magidor later changed to the method of (From|∼ to <BP ) in the context of “rational” systems of nonmonotonic reasoning, unfortunately

without commenting on this change.25For this reason, whenever PB( |∼ ) fails to be acyclic, so does PFH( |∼ ).26Suppose we interpret < abstractly as a relation that makes distinctions: A < B just

means that A and B can be told apart. Then the basic plausibility relation is coarser thanthe Friedman-Halpern plausibility relation. (But we should not put too much strain on thisinterpretation.)

17

Notice that for disjoint propositions A and B, φA <FHφB implies φA <BP

φB.The reason is that if φB ` ¬φA, then φA ∨ φB |∼φB implies φA ∨ φB |∼¬φA,by (RW), and that φA ∨ φB |∼φB and φA ∨ φB |6∼φA taken together implyφA ∨ φB |6∼¬φB, by (AND) and (RW). So the two relations <

BP= PB( |∼ )

and <FH

= PFH( |∼ ) derived from a given inference relation |∼ agree on allpairs of disjoint propositions.

6.2. The harmony between default reasoning and plausibility relations

The following observation makes sure that the Friedman-Halpern relation re-trieved from an inference relation for default reasoning |∼ is suitable for re-constructing this very relation. This result is, of course, due to Friedman andHalpern. Within this section, we suppose that |∼ satisfies (LLE), (REF),(AND), (RW) and (CP).

Observation 1 (Friedman and Halpern). Let |∼ be an inference relationand <

FH= PFH( |∼ ) be the Friedman-Halpern plausibility relation retrieved

from |∼ . Then the inference relation |∼ ′ = I(<FH

) is identical with |∼ . Inshort, I(PFH( |∼ )) = |∼ .

Proof. Let α |∼ ′β. Then by (From <FH

to |∼ ),

[[α ∧ ¬β]] <FH

[[α ∧ β]] or ∅ 6<FH

[[α]] .

Thus, by (From |∼ to <FH

)

(α ∧ β) ∨ (α ∧ ¬β) |∼α ∧ β and (α ∧ β) ∨ (α ∧ ¬β) |6∼α ∧ ¬β, or α |6∼α or α |∼⊥

That is, by (LLE) and (Ref)

α |∼α ∧ β and α |6∼α ∧ ¬β, or α |∼⊥

By the reflexivity condition (REF), (AND) and (RW), this means

α |∼β and α |6∼¬β, or α |∼⊥

Finally, by (AND) and (RW), this reduces to

α |∼β or α |∼⊥

and finally, by (RW) again, to

α |∼β

But this means that |∼ ′ is identical with |∼ . QED

Now we want to make sure that the basic plausibility relations retrieved froma default inference relation |∼ are in exactly the same way suitable for thereconstruction of |∼ as the Friedman-Halpern relations are. The following two

18

observations essentially reproduce results from Rott (2001, section 8.7) andRott (2003, section 3).

Observation 2. Let |∼ be an inference relation and <BP

= PB( |∼ ) be thebasic plausibility relation retrieved from |∼ . Then the inference relation |∼ ′ =I(<

BP) is identical with |∼ . In short, I(PB( |∼ )) = |∼ .

Proof. Let α |∼ ′β. Then by (From < to |∼ ),[[α ∧ ¬β]] <

BP[[α ∧ β]] , or [[α]] = ∅

Thus, by (From |∼ to <BP

),(α ∧ β) ∨ (α ∧ ¬β) |∼¬(α ∧ ¬β) and (α ∧ β) ∨ (α ∧ ¬β) |6∼¬(α ∧ β), or α ` ⊥that is,α |∼¬α ∨ β and α |6∼¬α ∨ ¬β, or α ` ⊥By the reflexivity condition (REF), (AND) and (RW), this is equivalent withα |∼β and α |6∼¬β, or α ` ⊥.By (REF), (AND), (RW) and (CP), this is equivalent withα |∼βBut this means that |∼ ′ is identical with |∼ . QED

Now we verify that we get a similar harmony result between basic plausibilityrelations <

BPand default inference relations |∼ if we start from the former

rather than from the latter.

Observation 3. Let <BP

be a basic plausibility relation and |∼ = I(<BP

)be the inference relation based on <

BP. Then the basic plausibility rela-

tion <BP′ = PB( |∼ ) retrieved from |∼ is identical with <

BP. In short,

PB(I(<BP

)) = <BP

.

Proof. Let A <BP′B. Then by (From |∼ to <

BP),

φA ∨ φB |∼¬φA and φA ∨ φB |6∼¬φBThus, by (From <

BPto |∼ )

(A∪B)∩−−A <BP

(A∪B)∩−A orA∪B = ∅, and (A∪B)∩−−B 6<BP

(A∪B)∩−Band A ∪B 6= ∅This condition reduces toA <

BP−A ∩B and B 6<

BPA ∩ −B and A ∪B 6= ∅

which, by (BP-iii) and (Asymmetry) is equivalent toA <

BPB, and A 6= ∅ or B 6= ∅

and by (Irreflexivity) this reduces toA <

BPB. QED

The same thing does not work equally well with <FH

.

Observation 4. Let <FH

be a Friedman-Halpern plausibility relation and|∼ = I(<

FH) be the inference relation based on <

FH. Then the Friedman-

Halpern plausibility relation <FH′ = PFH( |∼ ) is in general not identical with

<FH

. In short, generally PFH(I(<FH

)) 6= <FH

.

19

Proof. Let A <FH′B. Then by (From |∼ to <

FH),

φA ∨ φB |∼φB and φA ∨ φB |6∼φAThus, by (From <

FHto |∼ )

(A ∪B) ∩ −B <FH

(A ∪B) ∩B or ∅ 6<FHA ∪B, and

(A ∪B) ∩ −A 6<FH

(A ∪B) ∩A and ∅ <FHA ∪B

This condition reduces to(+) A ∩ −B <

FHB and −A ∩B 6<

FHA and ∅ <

FHA ∪B

This is not reducible toA <

FHB.

Both directions fail, as we will show with the help of two counterexamples. Letin the following W = {u, v, w} be the set of possible worlds.Example 2 above shows that (+) does not imply A <

FHB. Remember that in

this example the following plausibility values were assigned to the subsets of W :plaus(∅) = 0, plaus({u}) = 2, plaus({v}) = 1, plaus({w}) = 1, plaus({u, v}) =2, plaus({u,w}) = 2, plaus({v, w}) = 1 and plaus({u, v, w}) = 3. The plausi-bility relation < between subsets of W is again derived from these numbers byputting A < B iff plaus(A) < plaus(B). We verified before that (Qualitative-ness) is satisfied and thus we have an FH plausibility relation. But now considerA = {u, v} and B = {u,w}. We have A ∩ −B <

FHB and −A ∩ B 6<

FHA and

∅ <FHA ∪B, so (+) is satisfied, but not A <

FHB.

Example 4, which we are going to define now, shows that A <FHB does not

imply (+). Assign plausibility values to the subsets of W as follows (the onlyvalue that differs from Example 2 is that of {u,w}): plaus(∅) = 0, plaus({u}) =2, plaus({v}) = 1, plaus({w}) = 1, plaus({u, v}) = 2, plaus({u,w}) = 3,plaus({v, w}) = 1 and plaus({u, v, w}) = 3. The plausibility relation < betweensubsets of W is again derived from these numbers in the obvious way. The onlynon-trivial comparison for (Qualitativeness) concerns the three disjoint sets {u},{v} and {w}, and we can indeed verify that

{w} < {u} ∪ {v} and {v} < {u} ∪ {w} taken together imply {v} ∪ {w} < {u}

(This is 1 < 2 and 1 < 3 on the left-hand side, and 1 < 2 on the right-handside.) Thus the condition of Qualitativeness is satisfied here, we do in facthave an FH plausibility relation. Consider again A = {u, v} and B = {u,w}.Clearly, we have A <

FHB, but not −A ∩B 6<

FHA, violating (+). QED

7. Direct bridges between FH plausibilities and basic plausibilities

We can go directly from FH plausibilities to basic plausibilities, and back again.Given an FH plausibility relation <

FH, we define the corresponding basic plau-

sibility relation <BP

= PB(<FH

) by

(From <FH

to <BP

) A <BPB iff A <

FH−A ∩B

Conversely, given a basic plausibility relation <BP

, we define the correspondingFH plausibility relation <

FH= PFH(<

BP) by

20

(From <BP

to <FH

) A <FHB iff A ∩ −B <

BPB and −A ∩B 6<

BPA

Now it becomes clear that the meanings of plausibility judgements in Friedmanand Halpern’s language are in general different from those in the language ofbasic plausibility. For a correct understanding of the other camp’s statements,an interpreter is necessary. However, the meanings are not too far apart. Wecan show that in either translation, <

BPis a subrelation of <

FH, and as long FH

plausibilists and basic plausibilists restrict themselves to talking about disjointsets, no misunderstanding is possible, because they agree about such sets.

Lemma 5. (a) If <FH

is an FH plausibility relation, <BP

= PB(<FH

) andA <

BPB, then A <

FHB.

(b) If <BP

is a basic plausibility relation, <FH

= PFH(<BP

) and A <BPB,

then A <FHB.

(c) If <FH

is an FH plausibility relation, <BP

= PB(<FH

) and A and B aredisjoint propositions, then: A <

BPB iff A <

FHB.

(d) If <BP

is a basic plausibility relation, <FH

= PFH(<BP

) and A and B aredisjoint propositions, then: A <

FHB iff A <

BPB.

Proof. (a) Let <FH

be an FH plausibility relation, <BP

= PB(<FH

) andA <

BPB. By definition, this means that A <

FH−A∩B. So by FH-Dominance,

A <FHB.

(b) Let <BP

be a basic plausibility relation, <FH

= PFH(<BP

) and A <BPB.

We want to show that A <FHB, that is, by definition, A ∩ −B <

BPB and

−A ∩ B 6<BPA. But the former follows from A <

BPB by (Weak Continuing

down), (BP-vi). From A <BPB, we also get A <

BP−A ∩ B by (BP-iii). So

−A ∩B 6<BPA, by (Asymmetry), (BP-i), which is what we needed.

(c) Let A and B be disjoint propositions. Then −A ∩B is identical with B, so<

BP= PB(<

FH) agrees by definition with <

FHon disjoint propositions.

(d) Let A and B be disjoint propositions. Then A ∩ −B is identical with A,and −A ∩ B is identical with B. So the clause for the definition of A <

FHB

with <FH

= PFH(<BP

) reduces to A <BPB and B 6<

BPA. Since <

BPsatisfies

(Asymmetry), this reduces to A <BPB. Thus <

FH= PFH(<

BP) agrees with

<BP

on disjoint propositions. QED

Lemma 5 teaches an interesting lesson for plausibility-based default reasoning.The specific condition that (From < to |∼ ) uses for the definition of α |∼β refersto [[α ∧ β]] and [[α ∧ ¬β]] . Obviously, these propositions are disjoint. Lemma5 tells us that if we translate from the language of Friedman and Halpernto the language of basic plausibilities or vice versa, using either one of thedirect bridges PFH and PB between <

FHand <

BP, and then construct a

default inference relation using (From < to |∼ ), the result will be the sameon both sides. Although Friedman and Halpern have a different concept fromthe concept of a proponent of basic plausibility, the construction recipe both

21

parties use, viz., (From < to |∼ ), gives the same result for both. Within thespecific context of the construction of a plausibility-based inference relation, itdoes not matter that the meanings of the term “plausibility” differ.

Next we show that (Qualitativeness) of Friedman-Halpern relations correspondsroughly to the well-behavedness of basic plausibility relations.

Observation 5. If <BP

is a well-behaved basic plausibility relation, then<

FH= PFH(<

BP) is a qualitative FH-relation.

Proof. (FH0a) First we need to show (Irreflexivity).(FH0b) Next we show Transitivity.Let A <

FHB and B <

FHC, that is

(i) A ∩ −B <BPB and −A ∩B 6<

BPA

and(ii) B ∩ −C <

BPC and −B ∩ C 6<

BPB

In order to show that A <FHC, we need to show that

(iii) A ∩ −C <BPC and −A ∩ C 6<

BPA.

Let us first prove the first conjunct of (iii). From (i) and (ii) we get, using(BP-x),(A ∩ −B) ∪ (B ∩ −C) <

BPB ∪ C

By Meet right, (BP-iii), we get(A ∩ −B) ∪ (B ∩ −C) <

BP−((A ∩ −B) ∪ (B ∩ −C)) ∩ (B ∪ C)

or equivalently,(A ∩ −B) ∪ (B ∩ −C) <

BP(−A ∪B) ∩ (−B ∪ C) ∩ (B ∪ C)

The right-hand side is a subset of C, so we get, using (Continuing up),(A ∩ −B) ∪ (B ∩ −C) <

BPC

The left-hand side is a superset of A ∩ −C, so by (Continuing down),A ∩ −C <

BPC, which is the first conjunct of (iii).

Let us now turn to the second conjunct of (iii). Suppose for reductio thatalso −A ∩ C <

BPA. Taken together with A ∩ −C <

BPC, which we have just

established, this gives us, with the help of (BP-x),(A ∩ −C) ∪ (−A ∩ C) <

BPA ∪ C

Now we use Meet right, (BP-iii), and get(A ∩ −C) ∪ (−A ∩ C) <

BP−((A ∩ −C) ∪ (−A ∩ C)) ∩ (A ∪ C)

which is equivalent with(A ∩ −C) ∪ (−A ∩ C) <

BP(−A ∪ C) ∩ (A ∪ −C) ∩ (A ∪ C)

or simply(A ∩ −C) ∪ (−A ∩ C) <

BPA ∩ C

Now we join this with A ∩ −B <BPB from (i), using (BP-x) and get

(A ∩ −B) ∪ (A ∩ −C) ∪ (−A ∩ C) <BPB ∪ (A ∩ C)

Applying Meet right, (BP-iii) once more, we get(A∩−B)∪(A∩−C)∪(−A∩C) <

BP−((A∩−B)∪(A∩−C)∪(−A∩C))∩(B∪(A∩C))

which is equivalent with(A∩−B)∪(A∩−C)∪(−A∩C) <

BP((−A∪B)∩(−A∪C)∩(A∪−C))∩(B∪(A∩C))

The right-hand side of this inequality is a subset of B, so by (Continuing up),we get

22

(A ∩ −B) ∪ (A ∩ −C) ∪ (−A ∩ C) <BPB

The left-hand side of the inequality includes −B ∩C, so by (Continuing down),we infer−B ∩ C <

BPB

But this violates (ii). So we have found a contradiction and proved the secondconjunct of (iii). Therefore <

FH= PFH(<

BP) is transitive.

(FH1) Secondly, we need to show that<FH

= PFH(<BP

) satisfies FH-Dominance.So suppose that A ⊆ B.First, we show that if C <

FHA, then C <

FHB. That is, if C ∩ −A <

BPA and

−C ∩ A 6<BPC, then C ∩ −B <

BPB and −C ∩ B 6<

BPC. But C ∩ −A <

BPA

entails C ∩ −B <BPB due to (Continuing up) and (Continuing down), since

C ∩−B ⊆ C ∩−A. Similarly, −C ∩A 6<BPC and (Continuing down) entail that

−C ∩B 6<BPC.

Second, we show that if B <FHC, then A <

FHC. That is, if B∩−C <

BPC and

−B∩C 6<BPB, then A∩−C <

BPC and−A∩C 6<

BPA. But again, B∩−C <

BPC

implies A ∩ −C <BPC due to (Continuing down), and −B ∩ C 6<

BPB implies

−A ∩ C 6<BPA due to (Continuing up) and (Continuing down).

(FH2) Thirdly, we verify that <FH

= PFH(<BP

) satisfies (Qualitativeness).Since <

BPsatisfies the unrestricted condition (Choice hard) and <

FHagrees

with <BP

over disjoint sets, by Lemma 5(d), it follows immediately that <FH

satisfy Choice over disjoint sets, and that just means that it satisfies Qualita-tiveness.(FH3) Lastly, we need to show that ∅ <

FHA ∪ B implies that either ∅ <

FHA

or ∅ <FHB. Since <

FHagrees with <

BPover disjoint sets, by Lemma 5(d), it

is sufficient to show that ∅ <BPA ∪B implies that either ∅ <

BPA or ∅ <

BPB.

But the former implies that A ∪ B 6= ∅, by (Irreflexivity). Hence either A 6= ∅or B 6= ∅. So by (Minimality), either ∅ <

BPA or ∅ <

BPB. QED

Above (in subsection 4.4.1) we noted that the condition (Choice hard) doesnot follow from Friedman and Halpern’s condition (Qualitativeness), which isa restriction of (Choice hard) to disjoint propositions, even if (Irreflexivity),(Transitivity) and (FH-Dominance) are present as background conditions. Butif we construct basic plausibilities from FH plausibilities properly, that is, byusing the equation <

BP= PB(<

FH), we understand how the former can satisfy

the much stronger condition of (Choice hard), provided that latter satisfy theweaker condition (Qualitativeness).

The more general result is this:

Observation 6. If <FH

is a qualitative FH relation, then <BP

= PB(<FH

)satisfies the axioms for well-behaved basic plausibility relations except (Min-imality). If <

FHis a qualitative FH relation satisfying (Minimality), then

<BP

= PB(<FH

) is a well-behaved basic plausibility relation.

Proof. Let <FH

be a qualitative FH-relation and <BP

= PB(<FH

).(Irreflexivity). Suppose A <

BPA, that is, by definition, A <

FH−A ∩ A. This

means that A <FH∅, contradicting (FH1) and (Irreflexivity) for <

FH.

23

(Minimality). Let A 6= ∅. We need to show that ∅ <BPA, that is, by definition

∅ <FH−∅ ∩ A, which is ∅ <

FHA. This is guaranteed if, and only if, <

FH

satisfies (Minimality).(Continuing up). Let A <

BPB. By definition, this means that A <

FH−A∩B.

By (FH1), this implies A <FH−A∩ (B ∪C), which means, by definition again,

A <BPB ∪ C.

(Continuing down). Let A ∪ C <BPB. By definition, this means that A ∪

C <FH− (A ∪ C) ∩ B. By (FH1), this implies A <

FH−A ∩ B, which means,

by definition again, A <BPB.

(Choice easy) follows from (Continuing up) and (Continuing down).(Choice hard). Suppose that A <

BPB ∪C and B <

BPA∪C. We need to show

that A ∪B <BPC.

By the definition of PB(<FH

), our supposition means that(i) A <

FH−A ∩ (B ∪ C) and

(ii) B <FH−B ∩ (A ∪ C).

Condition (i) can be equivalently expressed as(i′) A <

FH(−A ∩B) ∪ (−A ∩ −B ∩ C)

Consider also the condition(ii′) −A ∩B <

FHA ∪ (−A ∩ −B ∩ C)

This condition is entailed by (ii), since −A∩B is a subset of B and −B∩(A∪C)is a subset of A ∪ (−A ∩ −B ∩ C), and <

FHsatisfies FH-Dominance.

Clearly, the sets A, −A ∩ B and −A ∩ −B ∩ C are pairwise disjoint. Now wecan apply the (Qualitativeness) of <

FHand conclude from (i′) and (ii′) that

(iii′) A ∪ (−A ∩B) <FH−A ∩ −B ∩ C

But A ∪ (−A ∩ B) is identical with A ∪ B, and −A ∩ −B ∩ C is identical with−(A ∪B) ∩ C, so (iii′) just means(iii) A ∪B <

FH−(A ∪B) ∩ C

By the definition of PB(<FH

), this just means that A∪B <BPC, which is what

we needed to show. QED

Due to the difference in the central concept of plausibility, it makes sense to saythat an advocate of basic plausibility speaks a language that is different fromthe language spoken by Friedman and Halpern. We can then view the bridgeprinciples between the two concepts of plausibility discussed in this paper astranslations. Translations are supposed to preserve meanings across differentlanguages. The only way to capture this idea in the present framework is totranslate back and forth and see, within a single language, whether there is anychange to what has been said. The result is that this idea works well in onedirection, starting from the language of basic plausibility, but it does not workin the other direction that starts from the language of Friedman and Halpern.

Observation 7. (a) Let <BP

be a basic plausibility relation and <BP′ =

PB(PFH(<BP

)). Then <BP′ = <

BP.

(b) Let <FH

be an FH plausibility relation and <FH′ = PFH(PB(<

FH)). Then

in general <FH′ 6= <

FH.

24

Proof. (a) Assume that A <BP′B. This means, by (From <

FHto <

BP),

A <FH−A ∩B

By Lemma 5(d), this is equivalent toA <

BP−A ∩B

which by (BP-iii) just meansA <

BPB

as desired.(b) Assume that <

FH′ = PFH(PB(<

FH)), and assume that

A <FH′B.

This means, by (From <BP

to <FH

),A ∩ −B <

BPB and −A ∩B 6<

BPA,

By Lemma 5(c), this is equivalent toA ∩ −B <

FHB and −A ∩B 6<

FHA.

By (FH1), the former conjunct of this is implied by A <FHB, and the latter

conjunct implies B 6<FHA. But this is all we can say. As counterexamples

against both directions we can use Example 2 and Example 4, exactly in thesame way as they are used in the proof of Observation 4. Thus there is noreduction to the target sentence A <

FHB. QED

Observation 7 shows that there is no perfect intertranslatability between thelanguage of Friedman and Halpern and the language of basic plausibility. Theobservation does not tell us, though, how this slightly disappointing result isto be interpreted. One direction works, the other does not, but where is theproblem? Is it due to the translation methods suggested in (From <

FHto <

BP)

or (From <BP

to <FH

)? I do not think so, since the methods have been sodesigned that PB(<

FH) = PB(I(<

FH)) and PFH(<

BP) = PFH(I(<

BP)). Is it

due to a defect in one of the theories? If we consider the design just mentionedand compare Observations 3 and 4, we find reason to say that the problemshowing up in the second part of Observation 7 is caused by the FH theoryrather than the theory of basic plausibility. This is not a very serious defect ofthe former theory, but it is not as harmoniously built up as the latter.

8. How to motivate the unrestricted Choice condition

It is nice that the properties of basic plausibility, in contrast to those of FHplausibility, can account for properties of the default reasoning operation in a(more or less) modular way. But doubts remain. The strong condition (Choicehard) is much stronger than the FH-condition (Qualitativeness). So shouldn’tit be much harder to motivate it? Fortunately, this is not the case. In fact Ithink it is fair to claim that it has been much better motivated than the latter.

The first steps for the motivation of the Choice conditions were already madein section 2. But this is only part of the story. Rott (2001, chapter 8) gives adetailed and systematic motivation.27 Here we have to be satisfied with a briefsketch of the argument. The whole approach is best understood in terms of

27This work also underlies the presentation of “basic entrenchment” in Rott (2003).

25

withdrawals (contractions) rather than revisions. This is the reason why (From.− to <

BE) features prominently in section 2.

The unrestricted Choice condition, i.e., the conjunction of (Choice easy) and(Choice hard), reads

A <BPB ∪ C and B <

BPA ∪ C if and only if A ∪B <

BPC

Using (<BP

and <BE

) and swapping negations, this can be transformed intotalk about entrenchments:

(Choiceent) B ∩ C <BEA and A ∩ C <

BEB if and only if C <

BEA ∩B

Concerning the left-hand side of (Choiceent), B ∩ C <BEA means

If one is to withdraw at least one of B ∩C and A, then one will withdrawB ∩ C and keep A

Because “withdrawing B ∩ C” means “withdrawing at least one of B and C”(see section 2), this in turn means:

If one is to withdraw at least one of A and B and C, then one will withdrawat least one of B and C and keep A.

Similarly, A ∩ C <BEB means

If one is to withdraw at least one of A and B and C, then one will withdrawat least one of A and C and keep B.

Concerning the right-hand side of (Choiceent), C <BEA ∩B means

If one is to withdraw at least one of C and A ∩B, then one will withdrawC and keep A ∩B

Because “withdrawing A ∩ B” means “withdrawing at least one of A and B”,and “keeping A ∩B” means “keeping both A and B”, this in turn means:

If one is to withdraw at least one of A and B and C, then one will withdrawC and keep both A and B.

This choice-theoretic interpretation shows that the left-hand side of (Choiceent)and the right-hand side of (Choiceent) mean exactly the same thing.

No such motivation is available for Friedman and Halpern’s condition of Qual-itativeness.

9. Conclusion

Friedman and Halpern’s work on plausibility measures is beautiful because itstarts from the very general conditions: Transitivity and Dominance (what we

26

called FH-Dominance). Being well below the level of preferential reasoning, thisset-up is suitable for covering both probabilistic and qualitative interpretations.By adding the condition of Qualitativeness, Friedman and Halpern get exactlywhat they declare to be the “core” of default reasoning, namely, preferentialreasoning.

A striking fact about the present author’s earlier work on basic entrenchmentis that the approach works in very general contexts—much more general thanAGM originally envisaged and, if suitably translated into the context of defaultreasoning, clearly more general than preferential reasoning. Basic plausibility,the concept introduced in the present paper, is the dual of basic entrenchment.It has been compared to (the relational perspective on) Friedman and Halpern’swork on plausibility measures.

The two approaches were developed independently (and roughly at the sametime), but have now turned out to be similar. Concerning the crucial and unfa-miliar conditions (Qualitativeness) and (Choice), some considerable differenceshave been identified. I have argued that the approach using basic plausibilitiesis preferable because it offers a more modular and better motivated account ofdefault reasoning.

A more general, and presumably more important, lesson from the exercise per-formed in this paper is that one can mean different things when speaking aboutplausibility in the context of default reasoning.

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